Explicit elastic curves
type: Item
instance of: scholarly article
title: Explicit elastic curves
zbMATH Open document ID: 0915.53003
DOI: 10.1023/A:1006526817291
author: Anders Linnér
published in: Annals of Global Analysis and Geometry
publication date: 1999-06-28T00:00:00Z
review text: The shape of a bent thin rod (a plane elastic curve) is given explicitly as soon as one knows certain three parameters. The basis of that is a theorem due to the author [cf. \textit{A. Linnér}, Nonlinear Anal., Theory Methods Appl. 21, 575-593 (1993; Zbl 0809.53006), J. Approximation Theory 84, 315-350 (1996; Zbl 0858.41004)], which relates the derivative (with respect to the arc length) of the tangent angle function to an elliptic function, containing three parameters. The parameters are to be found from geometric or elastic properties, in the worst case by nonlinear algebraic equations, but never by differential relations. Special choices of the elliptic functions and of the parameters yield a ``comprehensive, uniform, effective, and efficient'' (according to the author) collection of elastic curves. A number of examples illustrates, case by case, the full range of constraints (open and closed curves, with or without tension, without or with clamping at one or at both of the endpoints). From a differential geometric point of view, this is an interesting collection of plane curves, parametrized by arc length, whose geometric properties are given explicitly by elliptic functions.
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 74K10
zbMATH DE Number: 1211721
zbMATH Keywords: elastic curves in Euclidean space
zbMATH Keywords: mechanics of structures
zbMATH Keywords: rods
zbMATH Keywords: elliptic functions
reviewed by: E. Lanckau
MaRDI profile type: Publication
full work available at URL: a:1006526817291
OpenAlex ID: W1480905772
description: scientific article; zbMATH DE number 1211721